The Old Lady soon broke the deadlock when, in the 25th minute, Tevez picked up the ball, beat two opposition players and lashed a shot into the net past goalkeeper Antonio Mirante.
The Argentine, who used to play for Manchester United and Manchester City, got his second just seven minutes later to put him at the top of the Serie A scorers' standings.
Alessandro Lucarelli's effort just before the break, however, showed a signal of intent by the visitors.
Both sides exchanged shots at the start of the second half, but it was the Gialloblu who struck next, thanks to Molinaro's well-taken goal just after the hour.
Parma made things difficult for themselves, however, when Amauri got himself sent off just three minutes after his side's goal, his aerial clash with Giorgio Chiellini prompting the referee to show a red card.
Though just a goal behind, the away side, now reduced to 10 men, couldn't find the cutting edge in the final third to get a second, ensuring that the game, following a lengthy period of stoppage time, would gift Juventus all three points.